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10. Force Due to Liquid Pressure

by M. Bourne

The force F on an area A at a depth y in a liquid of density w is given by

F = wyA

The force will increase if the density increases, or if the depth increases or if the area increases.

So if we have an unevenly shaped plate submerged vertically in a liquid, the force on it will increase with depth. Also, if the shape of the plate changes as we go deeper, we have to allow for this.

So we have:

---

Now, the total force on the plate is given by

where

x is the length (in m) of the element of area (expressed in terms of y)

y is the depth (in m) of the element of area

w is the density of the liquid (in N m^-3)

(for water, this is w = 9800 N m^-3)

a is the depth at the top of the area in question (in m)

b is the depth at the bottom of the area in question (in m)

 

Example 1.

Find the force on one side of a cubical container 6.0 cm on an edge if the container is filled with mercury. The weight density of mercury is 133 kN/m³.

 

Answer

This is the same as having a square plate of sides 6.0 cm submerged in mercury.

 

This is a very basic example where the width of the plate does not change as we move down the plate.

It is always x = 6.

Also, the depth of the top of the plate is 0, so a = 0.

To apply the formula, we have:

x = 0.06 m

y = is the variable of integration

w = 133 kN m^-3 = 133000 N m^-3

a = 0

b = 0.06

So we have:

 

Example 2.

A right triangular plate of base 2.0 m and height 1.0 m is submerged vertically, with the top vertex 3.0 m below the surface.

Find the force on one side of the plate.

 

Answer

Before we can proceed, we need to find x in terms of y.

Now when x = 0, y = 3 and when x = 2, y = 4.

So we have:

This gives us x = 2y − 6.

To apply the formula, we have:

x = 2y − 6 = 2(y − 3)

y = is the variable of integration

w = 9800 N m^-3

a = 3

b = 4

So we have:

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Chapter Contents

• Applications of Integration
• 1. Applications of the Indefinite Integral
• 2. Area Under a Curve
• 3. Area Between 2 Curves
• 4. Volume of Solid of Integration
• 5. Centroid of an Area
• 6. Moments of Inertia
• 7. Work by a Variable Force
• 8. Electric Charges
• 9. Average Value
• Head Injury Criterion
• 10. Force by Liquid Pressure

• Comments, Questions?

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